Generic isometries and measure preserving homeomorphisms Christian - - PowerPoint PPT Presentation

Generic isometries and measure preserving homeomorphisms

Christian Rosendal University of Illinois at Chicago AMS-ASL Meeting Washington D.C. 2009

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

The main objects of our study are:

• continuous Lebesgue spaces, e.g., the unit interval [0, 1] with

Lebesgue measure λ or Cantor space 2N with Haar measure µ,

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

The main objects of our study are:

• continuous Lebesgue spaces, e.g., the unit interval [0, 1] with

Lebesgue measure λ or Cantor space 2N with Haar measure µ,

• the rational Urysohn metric space QU, which is the unique

(up to isometry) ultrahomogeneous, universal, countable metric space with distance set Q,

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

The main objects of our study are:

• continuous Lebesgue spaces, e.g., the unit interval [0, 1] with

Lebesgue measure λ or Cantor space 2N with Haar measure µ,

• the rational Urysohn metric space QU, which is the unique

(up to isometry) ultrahomogeneous, universal, countable metric space with distance set Q,

• the Urysohn metric space U, which is the unique

ultrahomogeneous, separable metric space, universal for all separable metric spaces.

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

However, the objects that really interest us are their groups of symmetries, namely Aut(2N, µ), the group of all measure-preserving Borel automorphisms of Cantor space,

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

However, the objects that really interest us are their groups of symmetries, namely Aut(2N, µ), the group of all measure-preserving Borel automorphisms of Cantor space, Homeo(2N, µ), the group of all measure-preserving homeomorphisms of Cantor space,

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

However, the objects that really interest us are their groups of symmetries, namely Aut(2N, µ), the group of all measure-preserving Borel automorphisms of Cantor space, Homeo(2N, µ), the group of all measure-preserving homeomorphisms of Cantor space, and Iso(QU) and Iso(U), the groups of isometries of these two spaces.

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

On Aut(2N, µ) we put the coarsest topology making all the mappings g → µ(g(A) △ B) continuous, where A and B run over all Borel subsets of 2N.

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

On Aut(2N, µ) we put the coarsest topology making all the mappings g → µ(g(A) △ B) continuous, where A and B run over all Borel subsets of 2N. On the other hand, when considering Homeo(2N, µ) we use a finer topology, namely such that all the sets {g ∈ Homeo(2N, µ)

• g(A) = B}

are open, where now A and B are clopen subsets of 2N.

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

Obviously, we have the following continuous inclusion of topological groups Homeo(2N, µ) ⊆ Aut(2N, µ). Moreover, in their respective topologies both groups are Polish, i.e., separable and completely metrisable.

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

Obviously, we have the following continuous inclusion of topological groups Homeo(2N, µ) ⊆ Aut(2N, µ). Moreover, in their respective topologies both groups are Polish, i.e., separable and completely metrisable. Therefore, we can talk about generic elements of these groups as those belonging to various comeagre subsets. Fact Homeo(2N, µ) is a dense subgroup of Aut(2N, µ).

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

Similarly, on Iso(U) we put the pointwise convergence topology, while on Iso(QU) we put the topology generated by the sets {g ∈ Iso(QU)

• g(x) = y}

for x, y ∈ QU.

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

Similarly, on Iso(U) we put the pointwise convergence topology, while on Iso(QU) we put the topology generated by the sets {g ∈ Iso(QU)

• g(x) = y}

for x, y ∈ QU. Again, these are Polish groups and, though less obvious than before, Fact Iso(QU) embeds continuously as a dense subgroup of Iso(U).

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

Theorem (Kechris–R. & Solecki & Rohlin & Kechris) The groups Homeo(2N, µ) and Iso(QU) have a comeagre conjugacy class.

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

Theorem (Kechris–R. & Solecki & Rohlin & Kechris) The groups Homeo(2N, µ) and Iso(QU) have a comeagre conjugacy class. On the other hand, all conjugacy classes in Aut(2N, µ) and Iso(U) are meagre.

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

Theorem (Kechris–R. & Solecki & Rohlin & Kechris) The groups Homeo(2N, µ) and Iso(QU) have a comeagre conjugacy class. On the other hand, all conjugacy classes in Aut(2N, µ) and Iso(U) are meagre. Actually, what is more important to us than the existence of comeagre conjugacy classes in the two first mentioned groups is the tool behind the proof of these facts. Theorem (Kechris–R. & Solecki) Homeo(2N, µ) and Iso(QU) are approximately compact, i.e., admit a countable increasing chain of compact subgroups with dense union.

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

Based on this latter result, we can now state our main result. Theorem The generic element g of Homeo(2N, µ) and of Iso(QU) is conjugate to all of its non-zero powers gn, n = 0.

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

Based on this latter result, we can now state our main result. Theorem The generic element g of Homeo(2N, µ) and of Iso(QU) is conjugate to all of its non-zero powers gn, n = 0. Explicitly, if g belongs to the comeagre conjugacy class of Homeo(2N, µ) or of Iso(QU), then so does gn for all n = 0.

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

Let us now see what this gives us.

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

Let us now see what this gives us. First of all, if g belongs to the comeagre conjugacy class Homeo(2N, µ) or of Iso(QU) and n = 0, then for some f we have g = fgnf −1 = (fgf −1)n, and so g has an nth root, which also belongs to the comeagre conjugacy class.

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

It follows that for any generic g we can inductively define g1, g2, g3, . . . such that g1 = g and for any n, (gn)n = gn−1.

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

It follows that for any generic g we can inductively define g1, g2, g3, . . . such that g1 = g and for any n, (gn)n = gn−1. This means that we can embed the additive group of Q into Homeo(2N, µ), respectively Iso(QU), by setting π( k n!) = gk

n

and π(1) = g.

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

It follows that for any generic g we can inductively define g1, g2, g3, . . . such that g1 = g and for any n, (gn)n = gn−1. This means that we can embed the additive group of Q into Homeo(2N, µ), respectively Iso(QU), by setting π( k n!) = gk

n

and π(1) = g. Corollary If g is a generic element of Homeo(2N, µ), then there is an action

• f the additive group Q by measure-preserving homeomorphisms
• n 2N such that 1 acts by g.

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

Using this corollary, we can deduce similar consequences for the

• ther groups.

Corollary (King) The generic measure-preserving automorphism has roots of all

• rders.

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms

Using this corollary, we can deduce similar consequences for the

• ther groups.

Corollary (King) The generic measure-preserving automorphism has roots of all

• rders.

Corollary The generic isometry of the Urysohn metric space U has roots of all orders.

Christian Rosendal University of Illinois at Chicago Generic isometries and measure preserving homeomorphisms